Modelling and analysis of a phytoplankton–zooplankton system with continuous and discrete time

Kartal, Şenol; Kar, Musa; Kartal, Neriman; Gürcan, Fuat

doi:10.1080/13873954.2016.1204323

Cited by 12 publications

(4 citation statements)

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“…Значительное продвижение в получении адекватной динамики при моделировании цветения фитопланктона достигнуто благодаря модификации классической системы взаимодействия фито и зоопланктона [27] путем использования уравнений с запаздыванием [например, [31][32][33], а также кусочно-непрерывных функций для учета дискретности в процессах интоксикации [17,35]. В частности, было показано, что природа цветения может быть описана колебаниями, возникающими в модели планктона в результате бифуркации Неймарка -Сакера, отражая периодичность цветения; причем возникновение этого явления зависит от скорости образования токсинов фитопланктоном, скорости роста зоопланктона и емкости экологической ниши.…”

Section: обсуждение и заключениеunclassified

“…Неудивительно, что огромное количество работ посвящено математическому моделированию планктонного сообщества. Простые модели, как правило, имеют в своей основе классические уравнения Лотки -Вольтерры [15], описывающие динамику системы «хищник -жертва», при этом количество модельных компонент экосистемы варьирует [напр., [16][17]. Наиболее популярны в моделировании планктонной экосистемы трехкомпонентные модели (т.н.…”

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“…Отдельные попытки учесть дискретность в процессах интоксикации предприняты в работах [17; 35]. В первой [17] предлагается модификация системы двух дифференциальных уравнений, описывающих динамику сообщества «фитопланктон -зоопланктон» [27] с использованием кусочно-постоянных аргументов. В ходе исследования авторам удалось показать, что природа цветения может быть описана колебаниями, возникающими в модели планктона в результате бифуркации Неймарка -Сакера, и тем самым продемонстрировать периодичность цветения.…”

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Discrete-Time Model of Seasonal Plankton Bloom

Неверова¹,

Zhdanova²,

Abakumov³

2020

Math.Biol.Bioinf.

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The most interesting results in modeling phytoplankton bloom were obtained based on a modification of the classical system of phytoplankton and zooplankton interaction. The modifications using delayed equations, as well as piecewise continuous functions with a delayed response to intoxication processes, made it possible to obtain adequate phytoplankton dynamics like in nature. This work develops a dynamic model of phytoplankton-zooplankton community consisting of two equations with discrete time. We use recurrent equations, which allows to describe delay in response naturally. The proposed model takes into account the phytoplankton toxicity and zooplankton response associated with phytoplankton toxicity. We use a discrete analogue of the Verhulst model to describe the dynamics of each of the species in the community under autoregulation processes. We use Holling-II type response function taking into account predator saturation to describe decrease in phytoplankton density due to its consumption by zooplankton. Growth and survival rates of zooplankton also depend on its feeding. Zooplankton mortality, caused by an increase in the toxic substances concentration with high density of zooplankton, is included in the limiting processes. An analytical and numerical study of the model proposed is made. The analysis shows that the stability loss of nontrivial fixed point corresponding to the coexistence of phytoplankton and zooplankton can occur through a cascade of period doubling bifurcations and according to the Neimark-Saker scenario leading to the appearance of quasiperiodic fluctuations as well. The proposed dynamic model of the phytoplankton and zooplankton community allows observing long-period oscillations, which is consistent with the results of field experiments. As well, the model have multistability areas, where a variation in initial conditions with the unchanged values of all model parameters can result in a shift of the current dynamic mode.

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Section: обсуждение и заключениеunclassified

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Discrete-Time Model of Seasonal Plankton Bloom

Неверова¹,

Zhdanova²,

Abakumov³

2020

Math.Biol.Bioinf.

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“…The populations models governed by differential equations have been studied extensively by many researchers (Chen et al, 2009;Feng and Kang, 2015;Lin, 2018;Wu et al, 2018) and the reference therein. However, in recent years considerable number articles in literature discussed the dynamic of the discretization of the models governed by differential equations (Kartal, 2016;Kartal et al, 2016;Din, 2018). Because the discrete-time models present dynamics consistency according to continuous-time.…”

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confidence: 99%

Stability and Period-Doubling Bifurcation in a Modified Commensal Symbiosis Model with Allee Effect

Işık

2022

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this article, the qualitative behaviour of discrete-time commensal symbiosis model which is obtained by implementing the forward Euler's scheme is discussed in detail. Firstly, the local stability conditions of fixed points of the model are studied. It is proved that the considered model undergoes Period-Doubling bifurcation around coexistence fixed point with the help of bifurcation theory. In order to support the accuracy of obtained analytical finding, some parameter values have been determined and numerical simulations are carried out for these parameter values. Numerical simulations display new and rich nonlinear dynamical behaviours. More specifically, when the parameter 𝛿𝛿 is chosen as a bifurcation parameter, it is seen that the considered discretetime commensal symbiosis model shows very rich nonlinear dynamical.

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1:2 Strong Resonance and Hybrid Control of Discrete Conformable Fractional Order Bacteria Population Model

Wang,

Deng

2024

Bull. Malays. Math. Sci. Soc.

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Modelling and analysis of a phytoplankton–zooplankton system with continuous and discrete time

Cited by 12 publications

References 40 publications

Discrete-Time Model of Seasonal Plankton Bloom

Discrete-Time Model of Seasonal Plankton Bloom

Stability and Period-Doubling Bifurcation in a Modified Commensal Symbiosis Model with Allee Effect

1:2 Strong Resonance and Hybrid Control of Discrete Conformable Fractional Order Bacteria Population Model

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